Math 103: Secants, Tangents and Derivatives
Ryan Blair
University of Pennsylvania
Thursday September 27, 2011
Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, 2011 1 / 11
Outline
1 Review
2 Secant Lines and Tangent Lines
3 Derivative
Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, 2011 2 / 11
Review
Limits Involving Infinity
1 Limits as x approaches ∞ or −∞.
2 Limits at infinite discontinuities
3 Horizontal asymptotes
4 Vertical asymptotes
Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, 2011 3 / 11
Review
Horizontal asymptotes
Definition
A line y = b is a horizontal asymptote of the graph of
y = f (x) if
limx→∞f (x) = b or limx→−∞f (x) = b
Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, 2011 4 / 11
Review
Horizontal asymptotes
Definition
A line y = b is a horizontal asymptote of the graph of
y = f (x) if
limx→∞f (x) = b or limx→−∞f (x) = b
Example: Find the horozontal asymptotes of
y = x+1+sin(x)x
Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, 2011 4 / 11
Secant Lines and Tangent Lines
Secant lines
Definition
A line in the plane is a secant line to a circle if it meets
the circle in exactly two points.
Definition
A line in the plane is a secant line to the graph of
y=f(x) if it meets the graph of y = f (x) in at least twopoints.
Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, 2011 5 / 11
Secant Lines and Tangent Lines
Secant lines
Definition
A line in the plane is a secant line to a circle if it meets
the circle in exactly two points.
Definition
A line in the plane is a secant line to the graph of
y=f(x) if it meets the graph of y = f (x) in at least twopoints.
Find the secant line to y = x3 − 2x + 1 between x = 0
and x = 1.
Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, 2011 5 / 11
Secant Lines and Tangent Lines
Tangent lines
Definition
A line in the plane is a tangent line to a circle if it
meets the circle in exactly one point.
Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, 2011 6 / 11
Secant Lines and Tangent Lines
Tangent lines
Definition
A line in the plane is a tangent line to a circle if it
meets the circle in exactly one point.
Definition
A line in the plane is a tangent line to the graph of
y=f(x) if it meets the graph of y = f (x) in exactly onepoint locally and it “kisses” the graph at that point
Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, 2011 6 / 11
Secant Lines and Tangent Lines
Tangent lines
Definition
A line in the plane is a tangent line to a circle if it
meets the circle in exactly one point.
Definition
A line in the plane is a tangent line to the graph of
y=f(x) if it meets the graph of y = f (x) in exactly onepoint locally and it “kisses” the graph at that point
“kisses” = (near the point of intersection the graph is
completely contain to one side of the tangent line)
Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, 2011 6 / 11
Secant Lines and Tangent Lines
Since we don’t have two points we need a new idea tofind the slope of a tangent line.
Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, 2011 7 / 11
Secant Lines and Tangent Lines
Since we don’t have two points we need a new idea tofind the slope of a tangent line.Find the slope of secant lines to f (x) = 1
1+xon the
following intervals:
1 [1, 3]
2 [1, 2]
3 [1, 1.5]
Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, 2011 7 / 11
Secant Lines and Tangent Lines
Since we don’t have two points we need a new idea tofind the slope of a tangent line.Find the slope of secant lines to f (x) = 1
1+xon the
following intervals:
1 [1, 3]
2 [1, 2]
3 [1, 1.5]
http://en.wikipedia.org/wiki/File:Sec2tan.gif
Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, 2011 7 / 11
Secant Lines and Tangent Lines
Since we don’t have two points we need a new idea tofind the slope of a tangent line.Find the slope of secant lines to f (x) = 1
1+xon the
following intervals:
1 [1, 3]
2 [1, 2]
3 [1, 1.5]
http://en.wikipedia.org/wiki/File:Sec2tan.gif
Tangent lines are the limits of secant lines
Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, 2011 7 / 11
Secant Lines and Tangent Lines
Tangent Lines
Definition
The tangent line to a curve y = f (x) at a point (a, f (a))
is the line through (a, f (a)) with the slope
limh→0f (a + h) − f (a)
h
Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, 2011 8 / 11
Secant Lines and Tangent Lines
Tangent Lines
Definition
The tangent line to a curve y = f (x) at a point (a, f (a))
is the line through (a, f (a)) with the slope
limh→0f (a + h) − f (a)
h
Find the slope of the line tangent to y = sin(x) at x = 0.
Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, 2011 8 / 11
Application: Instantaneous Velocity
Definition
If s(t) is a position function defined in terms of time t,then the instantaneous velocity at time t = a is given by
v(a) = limh→0s(a + h) − s(a)
h
Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, 2011 9 / 11
Application: Instantaneous Velocity
Definition
If s(t) is a position function defined in terms of time t,then the instantaneous velocity at time t = a is given by
v(a) = limh→0s(a + h) − s(a)
h
ExampleSuppose a penny is dropped from the top of
DRL which is 19.6 meters high. The position of the pennyin terms of hight above the street is given by
s(t) = 19.6− 4.9t2. At what speed is the penny travelingwhen it hits the ground.
Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, 2011 9 / 11
Derivative as a function
Given any function f (x) we want to find a new function
that, for any x-value, outputs the slope of f (x) at thatvalue.
Ryan Blair (U Penn) Math 103: Secants, Tangents and DerivativesThursday September 27, 2011 10 / 11
Derivative as a function
Given any function f (x) we want to find a new function
that, for any x-value, outputs the slope of f (x) at thatvalue.
Definition
f ′(x) = limh→0f (x + h) − f (x)
h
Ryan Blair (U Penn) Math 103: Secants, Tangents and DerivativesThursday September 27, 2011 10 / 11
Derivative as a function
Given any function f (x) we want to find a new function
that, for any x-value, outputs the slope of f (x) at thatvalue.
Definition
f ′(x) = limh→0f (x + h) − f (x)
h
Notation.Other ways of writing the derivative ofy = f (x).
f ′(x) = y ′ =dy
dx=
df
dx=
d
dxf (x) = Df (x) = Dx f (x)
Ryan Blair (U Penn) Math 103: Secants, Tangents and DerivativesThursday September 27, 2011 10 / 11
The Sandwich Theorem
Theorem
If f (x) ≤ g(x) ≤ h(x) when x is near c and
limx→cf (x) = limx→ch(x) = L
then limx→cg(x) = L
Ryan Blair (U Penn) Math 103: Secants, Tangents and DerivativesThursday September 27, 2011 11 / 11
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